A fast inversion technique for the interpretation of data from resistivity tomography surveys has been developed for operation on a microcomputer. This technique is based on the smoothness-constrained least-squares method and it produces a two-dimensional subsurface model from the apparent resistivity pseudosection. In the first iteration, a homogeneous earth model is used as the starting model for which the apparent resistivity partial derivative values can be calculated analytically. For subsequent iterations, a quasi-Newton method is used to estimate the partial derivatives which reduces the computer time and memory space required by about eight and twelve times, respectively, compared to the conventional least-squares method. Tests with a variety of computer models and data from field surveys show that this technique is insensitive to random noise and converges rapidly. This technique takes about one minute to invert a single data set on an 80486DX microcomputer.

Rapid least-squares inversion of apparent resistivity pseudosections by a quasi-Newton method1

Preface 1. Basic tools 2. Elasticity and Hooke's law 3. Seismic wave propagation 4. Effective media 5. Granular media 6. Fluid effects on wave propagation 7. Empirical relations 8. Flow and diffusion 9. Electrical properties Appendices.

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The Rock Physics Handbook: Tools for Seismic Analysis of Porous Media

Statistics and Data Analysis in Geology

Magnetotelluric (MT) data are inverted for smooth 2-D models using an extension of the existing 1-D algorithm, Occam’s inversion. Since an MT data set consists of a finite number of imprecise data, an infinity of solutions to the inverse problem exists. Fitting field or synthetic electromagnetic data as closely as possible results in theoretical models with a maximum amount of roughness, or structure. However, by relaxing the misfit criterion only a small amount, models which are maximally smooth may be generated. Smooth models are less likely to result in overinterpretation of the data and reflect the true resolving power of the MT method. The models are composed of a large number of rectangular prisms, each having a constant conductivity. Apriori information, in the form of boundary locations only or both boundary locations and conductivity, may be included, providing a powerful tool for improving the resolving power of the data. Joint inversion of TE and TM synthetic data generated from known models al...

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Occam's inversion to generate smooth, two-dimensional models from magnetotelluric data

Interpretation theory in applied geophysics: Grant, F S ... Interpretation theory in applied geophysics Unknown Binding – January 1, 1965 5.0 out of 5 stars 1 rating. See all formats and editions Hide other formats and editions. The Amazon Book Review Book recommendations, author interviews, editors' picks, and more. Read it now. Enter your mobile number or email address below and we'll send you a ...

Interpretation Theory in Applied Geophysics Grant (McGraw-Hill, 140s.)

Electromagnetic Theory for Geophysical Applications

The induced polarization (IP) and electromagnetic (EM) responses of a three‐dimensional body in the earth can be calculated using an integral equation solution. The problem is formulated by replacing the body by a volume of polarization or scattering current. The integral equation is reduced to a matrix equation, which is solved numerically for the electric field in the body. Then the electric and magnetic fields outside the inhomogeneity can be found by integrating the appropriate dyadic Green’s functions over the scattering current. Because half‐space Green’s functions are used, it is only necessary to solve for scattering currents in the body—not throughout the earth. Numerical results for a number of practical cases show, for example, that for moderate conductivity contrasts the dipole‐dipole IP response of a body five units in strike length approximates that of a two‐dimensional body. Moving an IP line off the center of a body produces an effect similar to that of increasing the depth. IP response va...

Three-Dimensional Induced Polarization and Electromagnetic Modeling

The electromagnetic fields scattered by a three‐dimensional (3-D) inhomogeneity in the earth are affected strongly by boundary charges. Boundary charges cause normalized electric field magnitudes, and thus tensor magnetotelluric (MT) apparent resistivities, to remain anomalous as frequency approaches zero. However, these E‐field distortions below certain frequencies are essentially in‐phase with the incident electric field. Moreover, normalized secondary magnetic field amplitudes over a body ultimately decline in proportion to the plane‐wave impedance of the layered host. It follows that tipper element magnitudes and all MT function phases become minimally affected at low frequencies by an inhomogeneity. Resistivity structure in nature is a collection of inhomogeneities of various scales, and the small structures in this collection can have MT responses as strong locally as those of the large structures. Hence, any telluric distortion in overlying small‐scale extraneous structure can be superimposed to ar...

Magnetotelluric responses of three-dimensional bodies in layered earths

We have developed a finite-difference solution for three-dimensional (3-D) transient electromagnetic problems. The solution steps Maxwell's equations in time using a staggered-grid technique. The time-stepping uses a modified version of the Du Fort-Frankel method which is explicit and always stable. Both conductivity and magnetic permeability can be functions of space, and the model geometry can be arbitrarily complicated. The solution provides both electric and magnetic field responses throughout the earth. Because it solves the coupled, first-order Maxwell's equations, the solution avoids approximating spatial derivatives of physical properties, and thus overcomes many related numerical difficulties. Moreover, since the divergence-free condition for the magnetic field is incorporated explicitly, the solution provides accurate results for the magnetic field at late times.An inhomogeneous Dirichlet boundary condition is imposed at the surface of the earth, while a homogeneous Dirichlet condition is employed along the subsurface boundaries. Numerical dispersion is alleviated by using an adaptive algorithm that uses a fourth-order difference method at early times and a second-order method at other times. Numerical checks against analytical, integral-equation, and spectral differential-difference solutions show that the solution provides accurate results.Execution time for a typical model is about 3.5 hours on an IBM 3090/600S computer for computing the field to 10 ms. That model contains 100 X 100 X 50 grid points representing about three million unknowns and possesses one vertical plane of symmetry, with the smallest grid spacing at 10 m and the highest resistivity at 100 Omega . m. The execution time indicates that the solution is computer intensive, but it is valuable in providing much-needed insight about TEM responses in complicated 3-D situations.

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A finite-difference, time-domain solution for three-dimensional electromagnetic modeling

We have developed an algorithm based on the method of integral equations to simulate the electromagnetic responses of three‐dimensional bodies in layered earths. The inhomogeneities are replaced by an equivalent current distribution which is approximated by pulse basis functions. A matrix equation is constructed using the electric tensor Green’s function appropriate to a layered earth, and it is solved for the vector current in each cell. Subsequently, scattered fields are found by integrating electric and magnetic tensor Green’s functions over the scattering currents. Efficient evaluation of the tensor Green’s functions is a major consideration in reducing computation time. We find that tabulation and interpolation of the six electric and five magnetic Hankel transforms defining the secondary Green’s functions is preferable to any direct Hankel transform calculation using linear filters. A comparison of responses over elongate three‐dimensional (3-D) bodies with responses over two‐dimensional (2-D) bodie...

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Gerald W. Hohmann

Papers

Electromagnetic Theory for Geophysical Applications

Three-Dimensional Induced Polarization and Electromagnetic Modeling

Magnetotelluric responses of three-dimensional bodies in layered earths

A finite-difference, time-domain solution for three-dimensional electromagnetic modeling

Electromagnetic modeling of three‐dimensional bodies in layered earths using integral equations